Talk:Triangular number

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Rotations in n-dimensional geometry

The amount of rotations in any n-dimensional geometry is the triangular number of n-1. Unfortunately, I haven't found anything on this on the internet so it would count as OR. Is there any way to get around this?

You can play around with it yourself if you want: https://www.mathgoodies.com/calculators/triangular-numbers

75.129.231.4 (talk) 23:19, 28 October 2020 (UTC)[reply]

What exactly do you mean by "amount of rotations?" Do you mean possible axes of rotation? In three dimensions an infinite number of rotational axes exist.—Anita5192 (talk) 00:00, 29 October 2020 (UTC)[reply]

Edward Waring

While looking at Edward Waring's Meditationes Algebraicae (3rd ed, 1782) for another purpose, I noticed that he states (in Latin, of course), without proof, that every whole number is the sum of 1, 2 or 3 triangular numbers. (This doesn't exclude the possibility that it is also the sum of 4, 5, 6... etc such numbers). (op. cit. p 349 prop. 3.) Waring goes on to state the corresponding propositions for pentagonal and hexagonal numbers (prop. 4) and squares (prop. 5). (This is on the same page as the proposition often known as Waring's Theorem.) I was wondering if this was already known before Waring. Looking at the article here, I see that it was later rediscovered by Gauss, who was excited enough to call it his Eureka theorem. However, I see that it is a special case of a conjecture of Fermat, known as his Polygonal Number Theorem, but characteristically stated without proof. Quite possibly Waring found it in Fermat. As a conjecture, without proof, it is therefore certainly earlier than Gauss, though Gauss may have been the first to prove it for the triangular case.2A00:23C8:7907:4B01:6415:B77:5411:4DE (talk) 14:06, 17 June 2022 (UTC)[reply]